that the foreign equity holding is 50 percent hedged. Note that these values suggest that the demand for lending in dollars and yen are as follows: d% = 10% dY = -10% Finally, the optimal portfolio allocations for a Japanese investor are given by the following weights: yu = 80% y} = 20% yx = 40% again implying that the foreign equity holding is 50 percent hedged. And note that these values imply that the demand for lending in dollars and yen are as follows: ys=-40% yY = 40% The reader may verify that the equilibrium conditions are satisfied. For example, the U.S. investors with 80 units of wealth demand 64 units of U.S. equities. Japanese investors with 20 units of wealth demand 16 units of U.S. equities, so that the total demand equals the total supply, a market capitalization weight of 80. How can one find these equilibrium values for the expected excess returns? One way would be to set up a simple algorithm that equates supply and demand. For example, in a spreadsheet you can define certain cells to have the various demands as a function of the expected excess returns. Then you can set other cells to be the excess demands, the difference between the demands and the supply, and ask the solver function to search for values of the expected excess returns that minimize the sum of squared excess demands. Such an approach will work in a simple example such as this, but it does not highlight the conditions that define an equilibrium. In order to accomplish this, in the next section we use matrix notation to show a more general approach. As should be clear from this simple two-country example, the international CAPM gets complicated very quickly. When we consider more than two countries, the notation and complexity of considering all the expected returns from various different points of view, the correlations and volatilities, and the relationships between them become cumbersome. In order to keep the notation as manageable as possible, in this section we use matrix algebra to simplify the presentation and develop the general approach.